تقرير
High-dimensional robust regression under heavy-tailed data: Asymptotics and Universality
| العنوان: | High-dimensional robust regression under heavy-tailed data: Asymptotics and Universality |
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| المؤلفون: | Adomaityte, Urte, Defilippis, Leonardo, Loureiro, Bruno, Sicuro, Gabriele |
| سنة النشر: | 2023 |
| المجموعة: | Computer Science Mathematics Condensed Matter Statistics |
| مصطلحات موضوعية: | Mathematics - Statistics Theory, Condensed Matter - Disordered Systems and Neural Networks, Computer Science - Machine Learning, Statistics - Machine Learning |
| الوصف: | We investigate the high-dimensional properties of robust regression estimators in the presence of heavy-tailed contamination of both the covariates and response functions. In particular, we provide a sharp asymptotic characterisation of M-estimators trained on a family of elliptical covariate and noise data distributions including cases where second and higher moments do not exist. We show that, despite being consistent, the Huber loss with optimally tuned location parameter $\delta$ is suboptimal in the high-dimensional regime in the presence of heavy-tailed noise, highlighting the necessity of further regularisation to achieve optimal performance. This result also uncovers the existence of a transition in $\delta$ as a function of the sample complexity and contamination. Moreover, we derive the decay rates for the excess risk of ridge regression. We show that, while it is both optimal and universal for covariate distributions with finite second moment, its decay rate can be considerably faster when the covariates' second moment does not exist. Finally, we show that our formulas readily generalise to a richer family of models and data distributions, such as generalised linear estimation with arbitrary convex regularisation trained on mixture models. Comment: 13 pages + Supplementary information |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2309.16476 |
| رقم الأكسشن: | edsarx.2309.16476 |
| قاعدة البيانات: | arXiv |
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